I need to complete some important questions for math, but I am unsure of how to complete them.
Here are the questions I need help with: 2. The parabola has the roots of ± √13 and passes through the point (-5, -4). Determine the equation of the parabola. Identify the vertex of the quadratic function.
3. Determine max/min value and when it happens using the indicated method.
(complete the square)
4. For what value(s) of k will the function f(x) = kx^2 -4x +k have:
(a) two equal real roots? (b) no zeros?
5. Visakan is making an arch at the top of his barn doors. The arch has a shape that can be modelled by the equation of a parabola. If the left edge of the door is the origin and the doorway is 6m wide, what is the equation of the parabola if the height of the arch is 2m from the edge of the doorway is 5m?
6. Stephen is throwing a horseshoe at a wooden stake. The function h(t) = -5t^2 +30t +1 models the height of the horseshoe in meters above the ground at time t. A tree 10m tall is between Stephen and the wooden stake.(a) What is the maximum height of the horseshoe? What is the minimum height?
(b) Will the horseshoe hit the tree? Explain your answer.
(c) About how long will it take for the horseshoe to reach the ground?
7. The demand function for a new perfume is p(x) = -2x +36 where p(x) represents the selling price, in thousands of dollars, and x is the number of bottles sold, in thousands.
a. Determine the revenue function and maximum revenue
b. At what price will the company receive maximum profits?
#2. y = a(x^2-13)
using the point, a(25-13) = -4, so a = -1/3
y = -1/3 (x^2-13)
so the vertex is at (0, 13/3)
#3. for y=ax^2+bx+c the vertex (max/min) is at x = -b/2a
#4. kx^2 - 4x + k has discriminant d=16-4k^2
two equal roots, d=0
no real roots, d<0
#5. the parabola is y=a(x-3)^2 + b
and we need y(2) = 5
Unfortunately, there are many parabolas that fit this, since you do not say how tall the door is.
#6. max height is at t = -b/2a = 3
so see whether that is at least 10.
for flight time, find when h=0
#7. revenue is price * quantity
now find the vertex of that parabola, as done above
2. To determine the equation of the parabola, we can use the vertex form of a quadratic equation: y = a(x-h)^2 + k, where (h, k) represents the coordinates of the vertex.
Since the parabola has the roots of ±√13, we know that the x-intercepts occur at x = -√13 and x = √13. This implies that the parabola is symmetric about the y-axis, so the x-coordinate of the vertex is 0.
Given that the parabola passes through the point (-5, -4), we can substitute these values into the vertex form equation to find the value of a:
-4 = a(0 - (-5))^2 + k
-4 = 25a + k
Now, we can substitute the value of a into the equation and solve for k:
-4 = 25a + k
-4 = 25(-4/25) + k
-4 = -4 + k
k = 0
Therefore, the equation of the parabola is y = -(4/25)x^2.
The vertex of the quadratic function is (0, 0).
3. To determine the maximum or minimum value of a quadratic function using the method of completing the square, we need to rewrite the quadratic equation in vertex form.
Let's consider a general quadratic equation: y = ax^2 + bx + c.
First, divide through by the coefficient of x^2 to make the coefficient equal to 1:
y = (a/b)x^2 + (b/b)x + (c/b)
Now, complete the square for the x-terms by adding and subtracting (b/(2a))^2:
y = (a/b)(x^2 + (b/a)x + (b/(2a))^2 - (b/(2a))^2 + (c/b)
Simplify the equation:
y = (a/b)(x + (b/(2a)))^2 + (c - (b/(2a))^2)
This equation is now in vertex form, y = a(x-h)^2 + k, where the vertex is at (h, k).
The maximum or minimum value occurs at the vertex, which is given by the coordinates (h, k).
4. To find the values of k that result in specific characteristics of the quadratic function f(x) = kx^2 - 4x + k:
(a) For the function to have two equal real roots, the discriminant should be zero. The discriminant is given by b^2 - 4ac, where a, b, and c are the coefficients of the quadratic equation.
In this case, a = k, b = -4, and c = k.
Substituting these values into the discriminant formula:
(-4)^2 - 4(k)(k) = 0
16 - 4k^2 = 0
4k^2 = 16
k^2 = 4
k = ±2
So, the function f(x) = 2x^2 - 4x + 2 or f(x) = -2x^2 - 4x - 2 will have two equal real roots.
(b) For the function to have no zeros, it must never cross or intersect the x-axis. This occurs when the discriminant is negative.
Using the discriminant formula as before:
(-4)^2 - 4(k)(k) < 0
16 - 4k^2 < 0
k^2 > 4
k > 2 or k < -2
Therefore, for any value of k greater than 2 or less than -2, the function will have no zeros.
5. To find the equation of the parabola that models the arch at the top of the barn doors, we need to determine the equation in vertex form.
Given that the left edge of the door is the origin (0,0) and the doorway is 6m wide, we know that the x-coordinate of the vertex is at (6/2) = 3.
Also, the height of the arch is 2m from the edge of the doorway (i.e., x = 5), so the y-coordinate of the vertex is at (2 + 0)/2 = 1.
Therefore, the vertex of the parabola is at (3, 1).
Using the vertex form equation, y = a(x-h)^2 + k, we can substitute these values into the equation:
y = a(x -3)^2 + 1
To find the value of the coefficient a, we can use another point on the parabola, such as (6, 0), which lies on the right edge of the doorway.
0 = a(6-3)^2 + 1
-1 = 9a
a = -1/9
Thus, the equation of the parabola is y = -(1/9)(x-3)^2 + 1.
6. (a) To find the maximum and minimum height of the horseshoe, we can first observe that the quadratic function h(t) = -5t^2 + 30t + 1 is in the form y = ax^2 + bx + c, where the coefficient of t^2 is negative, indicating a downward-opening parabola.
The maximum or minimum height occurs at the vertex of the parabola, which can be found using the formula h = -b/(2a).
For the given function, a = -5 and b = 30. Substituting these values into the formula:
h = -30/(2(-5))
h = 30/10
h = 3
Therefore, the maximum height of the horseshoe is 3 meters.
To find the minimum height, we can consider the fact that the coefficient of t^2 is negative, indicating that the parabola opens downwards and has no minimum point. Thus, the minimum height does not exist.
(b) To determine if the horseshoe will hit the tree, we need to find out whether the height of the horseshoe at any time is greater than 10 meters (height of the tree).
By analyzing the quadratic equation, we can see that the maximum height is 3 meters, which means that the horseshoe will not hit the tree as the maximum height is less than the height of the tree.
(c) The horseshoe will reach the ground when its height is equal to zero. We can set h(t) = 0 and solve for t:
-5t^2 + 30t + 1 = 0
We can solve this equation using factoring, quadratic formula, or completing the square. Assuming we use the quadratic formula:
t = (-b ± √(b^2 - 4ac))/(2a)
t = (-30 ± √(30^2 - 4(-5)(1)))/(2(-5))
t = (-30 ± √(900 + 20))/(-10)
t = (-30 ± √920)/(-10)
t = (-30 ± √4*230)/(-10)
t = (-30 ± 2√230)/(-10)
t = 3 ± (1/5)√230
Therefore, the horseshoe will take approximately 3 - (1/5)√230 seconds and 3 + (1/5)√230 seconds to reach the ground.
7. (a) The revenue function can be determined by multiplying the demand function p(x) by the number of bottles sold (x).
Revenue function = x * p(x)
Revenue function = x * (-2x + 36)
Revenue function = -2x^2 + 36x
To find the maximum revenue, we need to find the vertex of the revenue function using the formula x = -b/(2a).
For the revenue function, a = -2 and b = 36. Substituting these values into the formula:
x = -36/(2(-2))
x = -36/(-4)
x = 9
Therefore, the maximum revenue occurs when 9 thousand bottles of perfume are sold.
(b) To find the price at which the company will receive maximum profits, we need to identify the maximum point on the profit curve. The profit function can be determined by subtracting the cost function from the revenue function.
However, the cost function is not given in the problem statement, so we cannot determine the maximum price for maximum profits without that information.
Sure! I'd be happy to help you with your math questions. Let's go through each question one by one.
2. To determine the equation of the parabola, we need to use the fact that the roots are ±√13 and it passes through the point (-5, -4).
a. The roots give us two x-intercepts, so the equation can be written in factored form as follows:
(x - √13)(x + √13) = 0
b. Expanding this equation, we get:
x^2 - (√13)^2 = 0
x^2 - 13 = 0
c. Now, to find the equation of the parabola that passes through the point (-5, -4), we can substitute these coordinates into the equation:
(-5)^2 - 13 = 0
25 - 13 = 0
12 ≠ 0
d. Since 12 ≠ 0, the equation is not satisfied. Therefore, we need to introduce a multiplier "a" to the equation to fit the given point. Let's call the equation y = ax^2 + bx + c.
e. Substitute the coordinates of the point (-5, -4) into the equation:
-4 = a(-5)^2 + b(-5) + c
-4 = 25a - 5b + c
f. Also, substitute one of the roots of the parabola into the equation:
0 = a(√13)^2 + b(√13) + c
0 = 13a + √13b + c
g. With two equations and three unknowns (a, b, and c), we need one more equation. We can use the symmetry of the parabola to find the vertex.
h. The vertex of a parabola with equation y = ax^2 + bx + c is given by the formula:
x = -b / (2a)
i. Substitute this value of x back into the equation to find y:
y = a(-b / (2a))^2 + b(-b / (2a)) + c
j. Simplify:
y = (b^2) / (4a) - (b^2) / (2a) + c
k. To find the vertex, substitute the coordinates of the vertex, (-5, -4), into the equation:
-4 = (b^2) / (4a) - (b^2) / (2a) + c
With these equations, you can solve for the values of a, b, and c to determine the equation of the parabola as well as the vertex of the quadratic function.
I hope this helps you get started on your math questions! Let me know if you need further assistance.